How was the covenant of grace administered under the Old Testament?

Man by his fall having made himself incapable of life by that covenant, the Lord was pleased to make a second, commonly called the covenant of grace: wherein he freely offered unto sinners life and salvation by Jesus Christ, requiring of them faith in him that they may be saved, and promising to give unto all those that are ordained unto life his Holy Spirit, to make them willing and able to believe.

This covenant of grace is frequently set forth in the Scripture by the name of a testament, in reference to the death of Jesus Christ the testator, and to the everlasting inheritance, with all things belonging to it, therein bequeathed.

This covenant was differently administered in the time of the law and in the time of the gospel: under the law it was administered by promises, prophecies, sacrifices, circumcision, the paschal lamb, and other types and ordinances delivered to the people of the Jews, all fore-signifying Christ to come, which were for that time sufficient and efficacious, through the operation of the Spirit, to instruct and build up the elect in faith in the promised Messiah, by whom they had full remission of sins and eternal salvation; and is called the Old Testament.

Under the gospel, when Christ the substance was exhibited, the ordinances in which this covenant is dispensed are the preaching of the Word and the administration of the sacraments of Baptism and the Lord's Supper; which, though fewer in number, and administered with more simplicity and less outward glory, yet in them it is held forth in more fulness, evidence, and spiritual efficacy, to all nations, both Jews and Gentiles; and is called the New Testament. There are not, therefore, two covenants of grace differing in substance, but one and the same under various dispensations.

This perseverance of the saints depends, not upon their own free will, but upon the immutability of the decree of election, flowing from the free and unchangeable love of God the Father; upon the efficacy of the merit and intercession of Jesus Christ; the abiding of the Spirit and of the seed of God within them; and the nature of the covenant of grace: from all which ariseth also the certainty and infallibility thereof.

Sacraments are holy signs and seals of the covenant of grace, immediately instituted by God, to represent Christ and his benefits, and to confirm our interest in him: as also to put a visible difference between those that belong unto the Church and the rest of the world; and solemnly to engage them to the service of God in Christ, according to his Word.

Baptism is a sacrament of the New Testament, ordained by Jesus Christ, not only for the solemn admission of the party baptized into the visible Church, but also to be unto him a sign and seal of the covenant of grace, of his ingrafting into Christ, of regeneration, of remission of sins, and of his giving up unto God, through Jesus Christ, to walk in newness of life: which sacrament is, by Christ's own appointment, to be continued in his Church until the end of the world.